San Jose Math Circle
February 7, 2009
CIRCLES AND FUNKY AREAS - PART I
DEFINITIONS:
• The set of all points that are the same distance from a given point is a circle. The
given point is the center of the circle, and the fixed distance is the radius.
• We often refer to a circle by its center using the symbol ; O refers to a circle with
center O.
• A line that touches a circle in a single point is tangent to the circle, while a line that
hits a circle in two points is a secant. A segment connecting two points on a circle is
a chord, and a chord that passes through the center of the circle is a diameter.
• The portion of a circle that connects two points on the circle is an arc, which we denote
_
with the endpoints of the arc: M N is the shorter arc that connects M and N .
• The perimeter of a circle is called the circle’s circumference.
• A central angle of a circle is an angle whose vertex is at the center at the circle.
• A portion of a circle cut out by drawing two radii of the circle is called a sector of the
circle.
• A portion of a circle between a chord and the arc of the circle connecting the endpoints
of the chord is a circular segment of the circle.
• In every single circle, the ratio of circumference to diameter is the same. This ratio is
called pi, and is given the symbol π. Its value is approximately 3.14. Pi is an irrational
number, which means that is cannot be expressed as a ratio of integers. Because pi is
irrational, its decimal expansion does not terminate and does not become periodic.
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FUN FACTS:
• 22/7 is a common approximation of π. 355/113 is an even better approximation. In 1914, the great
Indian mathematician Ramanujan provided the uncanny approximation:
r
192
4
π ≈ 92 +
22
•
1
1
1
1
1
π2
= 1 + 2 + 2 + 2 + 2 + ...
6
1
2
3
4
5
• In 1897, the Indiana state legislature almost passed a bill that set the value of π to exactly 3.2. The
House voted unanimously for it and it passed a first reading in the Senate. Fortunately, a math
professor at Purdue University happened to be visiting the legislature at the same time and advised
that the bill be postponed indefinitely, effectively killing it.
FORMULAS:
Let O have radius r and let AB be a chord with ∠AOB = x◦ .
_
• The degree measure of the arc AB is:
_
AB = x◦
• The circumference C of the circle is:
C = 2πr
_
• The length of the arc AB is:
_
AB =
2
x
· 2πr
360
• The area A of the circle is:
A = πr2
• The area of the sector AOB is:
Area sector AOB =
x
· πr2
360
REMARK. If the angle ^AOB is given in radians and m(^AOB) = θ radians, then the
_
formulas for calculating the length of the arc AB and the area of the sector AOB become:
_
• Length of the arc AB:
_
AB = rθ
• Area of the sector AOB:
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Area sector AOB = r2 θ
2
3
1. (Harvard-MIT Mathematics Tournament) A man standing on a lawn is wearing a circular sombrero of radius 3 feet. Unfortunately, the hat blocks the sunlight so effectively
that the grass directly under it dies instantly. If the man walks in a circle of radius 5
feet, what area of dead grass will result?
2. (AOPS) When my car has wheels with a diameter of 24 inches, its speedometer reports
the correct speed of my car. I recently replaced my 24-inch wheels with 28-inch diameter wheels. I didn’t change my speedometer, however. When the speedometer tells
me that the car is going 40 miles per hour (and I’m driving with my 28-inch wheels),
how fast is my car really going?
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3. (MATHCOUNTS) Two congruent circular coins A and Z are touching at point P . A
is held stationary while Z is rolled around it one time in such a way that the two coins
remain tangent at all times. How many times will Z revolve around its center?
4. (AOPS) Regular hexagon ABCDEF is inscribed in O with radius 6. What is the
ratio of the circumference of the circle to the perimeter of the hexagon?
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5. (MATHCOUNTS) On each side of a right triangle a semicircle is constructed using
that side as a diameter. How many square centimeters are in the area of the semicircle
on the hypotenuse of the right triangle if the areas of the semicircles on the legs of the
triangle are 36 and 64 square centimeters?
6. (MATHCOUNTS) What is the number of square centimeters in the area that is not
shaded in the diagram below? The radius of the large semicircle is 1 centimeter, the
radius of the small circle is 0.5 centimeters, and the length of the longer leg on the
right triangle is 3 centimeters.
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7. (ARML) The larger circle at right has radius 1.5 times the smaller circle. Compute
the ratio of the partial ring ABCD to the area of sector BOC.
8. (2004 AMC 10A, ]21) Two distinct lines pass through the center of three concentric
circles of radii 3, 2, and 1. The area of the shaded region in the diagram is 8/13 of the
area of the unshaded region. What is the radian measure of the acute angle formed by
the two lines? (Note: π radians is 180 degrees).
(A)
π
8
(B)
π
7
(C)
π
6
(D)
π
5
(E)
7
π
4
9. (2000 AMC 10, ]18) Charlyn walks completely around the boundary of a square whose
sides are exactly 5 km long. From any point on her path she can see exactly 1 km
horizontally in all directions. What is the area of the region consisting of all points
Charlyn can see during her walk, expressed in square kilometers and rounded to the
nearest whole number?
(A) 24
(B) 27
(C) 39
(D) 40
(E) 42
10. (2003 AMC 10 A, ]17) The number of inches in the perimeter of an equilateral triangle
equals the number of square inches in the area of its circumscribed circle. What is the
radius, in inches, of the circle?
√
√
√
√
3 3
6
3 2
(B)
(C) 3 (D)
(E) 3π
(A)
π
π
π
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11. (AOPS) O is the center of the shown circle and OA = 8. The shaded region between
chord AB and the circle is called a circular segment. Find the area of this circular
segment.
12. (AOPS) Find the area of the shaded region given that O is the center of the circle,
∠AOB = 120◦ , and the radius of the circle is 6.
13. (AOPS)√Given that 4ABD in the diagram below is a right triangle with BD = 8 and
AB = 8 3, find the total area of the shaded regions.
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14. (AOPS) Each side of equilateral triangle XY Z has length 9. Find the area of the
region inside the circumcircle of the triangle, but outside the triangle.
15. (AOPS) I have a barn that is a regular hexagon, as shown. Each side of the barn is
100 feet long. I tether my burro to point A with a 150 foot rope. Find the area of the
region in which my burro can graze.
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16. (AOPS)√ The shaded portion of the figure is called a lune. Given that AB = 1,
CD = 2, and that AB and CD are diameters of the respective semicircles shown,
find the area of the lune.
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